| import OAI.RingTheory.DirectFiniteness.GluedGroup |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped Classical |
| open scoped BigOperators |
| open Equiv Equiv.Perm |
| namespace KaplanskyCounterexample.Planar |
| variable {V F A B X : Type} [AddCommGroup A] [AddCommGroup B] |
| variable {Inc : V → F → Prop} {ι : V → F → (A →+ B)} {f₀ : F} |
|
|
| /-- Insertions and deletions preserve the existence of the anchored lifetime drawing. -/ |
| theorem OpenDiagram.related_iff (token : X × Bool → F × A) |
| (Allowed : List (X × Bool) → Prop) |
| (hA : ∀ r, Allowed r → ∃ v, (∀ t ∈ r.map token, Inc v t.1) ∧ |
| ((r.map token).map (fun t => ι v t.1 t.2)).sum = 0) |
| {u w : List (X × Bool)} (h : WordReduction.Related Allowed u w) : |
| Nonempty (OpenDiagram Inc ι f₀ (u.map token)) ↔ |
| Nonempty (OpenDiagram Inc ι f₀ (w.map token)) := by |
| induction h with |
| | rel u w h => |
| cases h with |
| | block pre r post hr => |
| obtain ⟨v, hi, hs⟩ := hA r hr |
| simp only [List.map_append] |
| constructor |
| · rintro ⟨D⟩ |
| exact D.deleteWordBlock _ _ _ v hi hs |
| · rintro ⟨D⟩ |
| exact D.insertWordBlock _ _ _ v hi hs |
| | refl => rfl |
| | symm _ _ _ ih => exact ih.symm |
| | trans _ _ _ _ _ ih₁ ih₂ => exact ih₁.trans ih₂ |
|
|
| end KaplanskyCounterexample.Planar |
|
|
| end |
|
|
| end OAI |
|
|
